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ps72003 - ACM 95b/100b Problem Set 7 Due March 7 2003 3pm...

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Problem Set 7 February 28, 2003 Due March 7, 2003 ACM 95b/100b 3pm at Firestone 303 (2 pts) Include grading section number Read: For series solutions, see BdP Chapter 5 and also look at Prof. Pierce’s online notes. For separation of variables, read BdP Section 10.5; or Haberman Chapter 1, Sections 2.1–2.3; or Farlow Lessons 1–5. 1. (5 points each) For each of the following partial differential equations, what system of ordinary dif- ferential equations results when separation of variables is applied? (Make sure to clearly define your separation constants.) a) u tt = c 2 u xx one-dimensional wave equation, c =wave speed b) u xx + u yy + u zz = 0 Laplace’s equation in Cartesian 3-space c) u t = ku xx cu x one-dimensional advection-diffusion equation ( k =diffusion coefficient, c =advection speed) d) v t + rxv x + 1 2 σ 2 x 2 v xx = rv Black-Scholes PDE v ( x, t )=stock option price, x =price of the option’s underlying stock, t =time Constants: r =interest rate, σ = volatility of the underlying stock 2. (5 points each) What systems of ordinary differential equations arise when separation of variables is applied to the heat equation u t = κ
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  • Winter '09
  • NilesA.Pierce
  • Partial differential equation, heat equation ut, one-dimensional wave equation, one-dimensional advection-diffusion equation, r2 ∂r r2

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